Math at Work 11: Chapter 2 - Nelsonlearningcentre.nelson.com/.../SE/02MW11_Chapter2_Fin.pdf2.1...

Math at Work 11: Chapter 2February 20, 2012, 16:15

George builds model dories. He is meticulous with the

accuracy of his measurements. His creations are exact

scale replicas of Newfoundland Dories.

1. What is a scale replica?

2. Describe some scale replicas you own or have seen.

3. What measurements do you think George needs to

make in order to build a model to scale?

Key Words

scale

scale drawing

orthographic drawing

isometric drawing

one-point perspective drawing

vanishing point

point of perspective

exploded view diagram

56 MHR • Chapter 2

Career LinkBryan is a framer. He builds walls, door

frames, stairs, and roof trusses, among

other things. Bryan uses other people’s

drawings to determine the size and

location of the items he is responsible

for building.

Drawing and Design • MHR 57

Get Ready

Proportional Thinking

1. Solve to create equivalent fractions.

a) 1_4

=�

_8

b) 1_8

=�

_16

c) 1_4

=�

_400

d) 1_12

=�

_60

e) 1_4

= 8_�

f) 1_2

= 16_�

g) 1_10

=70_�

h) 1_20

=40_�

2. Solve to create equivalent ratios.

a) 1 : 5 = � : 20

b) 1 : 12 = � : 72

c) 2 : 3 = � : 12

d) 9 : 16 = � : 48

e) 40 : 1 = 200 : �

f) 8 : 5 = 200 : �

3. 7 ft

5 ft

a) What is the perimeter of the

rectangle?

b) What is the area of the rectangle?

c) If the length of the rectangle is

doubled, is the perimeter doubled?

d) If the length of the rectangle is

doubled, is the area doubled?

e) Calculate the perimeter and the

area of the larger rectangle.

Working With Diagrams

These are examples of hash marks: and .

Hash marks on a diagram identify sides that

are the same length.

4. Solve for the unknown values.

a)

40 cm

80 cm �

b) 4.2 m

9 m �

�c)

9�

�

� �

d)

4��

�

5. For each right triangle, determine the

missing length.

To determine the missing length of a

right triangle, use the Pythagorean

relationship.

a2 + b2 = c2

42 + b2 = 52

16 + b2 = 25

b2 = 9

b = √ __

9

b = 3

The length of b is 3 cm.

a = 4

bc = 5

a

17 in. 15 in.

b)

5.6 cm

3.3 cm c


Area

6. Determine the area of each shape.

a)

7 ft

3 ft

b)

3.6 m

7 m

7. Calculate each area, to the nearest

square metre.

a)

r = 7 m

b)

d = 12 m

8. Calculate the area of the composite

shape.

12 ft

8 ft

8 ft

Nets and Surface Area

9. Name the shape made by each net.

a)

b)

10. Estimate and then determine the

surface area of each fi gure.

a)

2.4 m

2.2 m

5.2 m

b) d = 10 cm

10 cm

11. On grid paper, sketch a net of each of

the fi gures in #10.

Get Ready • MHR 59

Working With Scale

A hobbyist building a model, a carpenter reading a drawing,

and a dressmaker working from a pattern all need to have a

good knowledge of scale. Also, long-distance truck drivers

who must interpret the scale of a map.

Th e Sopwith Camel was a British World War I

fi ghter plane. Th is picture shows

a scale model of the Sopwith

Camel. In a scale model,

every dimension of the

actual plane has been

reduced by the same

factor.

scale

the relationship • between a distance

on a drawing, model,

or map and the

actual distance

for example, a scale • of 1 cm : 1 m means

that 1 cm on the

diagram, model, or

map represents 1 m

actual size

Explore Scale

wingspan

Focus On . . .

using proportional • reasoning to determine

the dimensions of a scale

drawing or model

drawing a scale diagram • of a given object

solving problems that • involve scale

describing when a scale • representation might

be used

constructing a model of a • 3-D object


Th e actual wingspan of the Sopwith Camel is 28 feet.

1. Convert 28 feet to inches.

2. Measure the distance of the wingspan in the picture, in inches.

3. How many times greater is the actual wingspan than the one in

the picture?

4. Refl ect

a) Express the wingspan in the picture and

the wingspan of the actual plane as a ratio.

b) State the ratio in its simplest form.

5. Extend Your Understanding Measure the diameter of one of

the wheels in the picture. Use the ratio from step 4b) to determine

the approximate diameter of the wheel on a real Sopwith Camel.

Model builders produce life-size and scale models of many vintage

planes, including the Sopwith Camel and the Vickers Vimy, shown here.

A replica of the Vickers Vimy made a non-stop fl ight from St. John’s, NL,

to Ireland in 2005. This trip duplicated early fl ights made by the

original Vickers Vimy aircraft. To make replicas, model builders need

precision measuring tools and a knowledge of scale. To learn more

about vintage aircraft models and the people who make them, go to

www.mcgrawhill.ca/school/learningcentres and follow the links.

Tools of the Trade

Materials

imperial ruler•

Express the ratio

in the form � : �.

Web Link

Plan a scale model

of the solar system.

Decide on the

diameter of the sun

for your scale model,

and learn what sizes

the planets should

be. Go to www.

mcgrawhill.ca/school/

learningcentres and

follow the links.

2.1 Working With Scale • MHR 61

Read Scale Drawings

Th is is a scale drawing of Fiona’s bedroom done on 1 _ 4

inch grid paper.

bedroom

door

door

window

closet

Scale: 1 square represents 6 in.

a) Explain the scale.

b) Convert the scale of the diagram to a 1 : � ratio.

c) What are the dimensions of Fiona’s room, in feet?

d) How wide are the doors? Door dimensions are quoted in inches.

Show your answer in inches.

e) How deep is the closet? Show your answer in feet and inches.

Solution

a)

On the Job 1

scale drawing

two-dimensional • (2-D) drawing used to

represent a place or

object

uses scale to show • the relationship

between the distance

on a drawing and the

distance in real life

A ratio shows the

relationship between

two values that have

the same units. A

scale can show the

relationship between

two values that have

the same units or

diff erent units.


b) Th e diagram is on 1 _ 4

inch grid paper. Th e length of one side of each

square on the diagram is 1 _ 4

� long.

1 _ 4

� represents 6�.

2 _ 4

� represents 12�.

3 _

4 � represents 18�.

1� represents 24�.

Th e drawing is a 1 : 24 reduction of the room.

c) On the diagram, the length of the room is 22 squares.

Since 2 squares represent 1 foot, the actual room is

22 ÷ 2 = 11 feet long.

Th e width of the room is 18 squares, so the room is 9 feet wide.

d) One square represent 6 inches. Th e closet opening is 4 squares, so

the closet door is 4 × 6 = 24 inches wide.

On the diagram, the main door to the room is 5 squares wide.

Th e door to the room is 5 × 6 = 30 inches wide.

e) Th e depth of the closet is 6 1 _ 2

squares on the diagram.

6.5 × 6 = 39

Th e closet is 39 inches deep.

Change 39 inches to feet and inches.

1′ = 12�

2′ = 24�

3′ = 36�

39� = 3′ 3�

Th e closet is 3′ 3� deep.

Your Turn

Refer back to the scale drawing of the bedroom.

a) How far from the top-left corner of the room is the middle of

the window? State the measurement in inches. Hint: Use the

Pythagorean relationship.

b) How wide is the window? State the measurement in feet.

c) How wide is the closet? State the measurement in feet and inches.

Look for a

Pattern

Strategy

Ratios can also be

writt en as fractions.

1 : 24 can also be

writt en as 1

_ 24

.

6 × 6 = 36

0.5 × 6 = 3

36 + 3 = 39

Look for a

Pattern

Strategy


Try It 1. Jackie is building a scale model of a garden shed. She will let 1 inch

represent 2 feet. If the base of the shed measures 8 feet by 12 feet,

what measurements will Jackie need for the model?

2. Write each scale as a 1 : � ratio.

a) 1 cm to 1 m b) 1 inch to 1 foot

c) 1 cm to 1 km d) 1 _ 4

in. to 1 ft

3. A common scale for collectible

toy cars is 1 : 64. Use the

following measurements of

the scale model to

determine the actual

measurements of the 1959

Volkswagen Beetle. Round

your answers to the nearest

centimetre.

a) length = 6.4 cm

b) width = 2.4 cm

c) height = 2.3 cm

d) wheel diameter = 5.6 mm

4. Th e most famous aircraft designed and built in Canada was the

Avro Arrow. A number of 1 _ 8

scale models were made and tested in

a wind tunnel. Th e length of each model was 10′ 8�. What was the

length of the full-size Arrow?

Check Your Understanding

Web Link

Check the size of an

actual Volkswagen

Beetle by going to

www.mcgrawhill.ca/

school/learningcentres

and following the

links.

Engineers

frequently start

with scale models

of the items they

plan to build. Scale

models allow them

to test how the

environment will

aff ect the item. To

learn more about

building models

for this purpose, go

to www.mcgrawhill.ca/


and follow the links.

Tools of the Trade


Apply It 5. Part of a scale drawing for

Josh’s landscape design is

shown below. Th e design

is drawn on 1 _ 4

inch grid

paper.

For parts a) to c), measure

from the centre of the tree.

a) How far from the

steps does Josh plan

to plant tree #1?

b) How far from the

deck will he plant tree #2?

c) Calculate the distance between the two trees, to the nearest

foot. Hint: Use the Pythagorean relationship.

d) Using a fl oor with square tiles, mark the location of the two

trees relative to each other.

6. Create a scale drawing of a room in your school. Choose an

appropriate scale. Include the locations of all doors and windows.

7. Determine the number of 6�-by-6� fl oor tiles needed to cover the

entrance and lobby of this recreation centre. Th e entrance and

lobby are shaded blue.

office

Scale: 1 square represents 1′

entrance

lobby

washroom

to gym

Japanesemaple

deck

#1

#2

Scale: � represents 1′1–4


Determine Missing DimensionsTo avoid clutter, many scale drawings include a minimum amount of

information. Use the measurements given to determine any missing

dimensions.

Determine the lengths of walls A to E identifi ed on the diagram.

B

AC

E

D12′

24′

9′

On the Job 2


Solution

Th e fi rst dimension that can be determined is D.

It is the same length as the wall with the same number of hash marks.

So, wall D is 9 feet long.

B is next. 9 + 9 = 18. Th e length of the wall at the bottom of the

drawing is 24 feet. Th e length of the wall at the top must be the same.

Th erefore, B must be 24 - 18 = 6 feet long.

Th e hash marks on the diagram indicate that A and C are the same

length as B, so they are 6 feet long as well.

Since A is 6 feet and the

overall dimension on

the left of the diagram

is 12 feet, E must also

be 6 feet long.

Your Turn

Determine the missing dimensions.

a)

6 m

17 m

4 m

Y

X12 m

b)

8′

P

R

T

Q S

U

V

12′

18′

C is also 6′ long. You could

use the length of C to

calculate the length of E.


Try It 1. Determine the missing dimensions.

a) 8.2 cm

BA

38 cm

24.5 cm

13.6 cm

10.4 cm

b) 10�

6�

10�

D

C

3 �12

2 �14

2. Calculate the perimeters of the fi gures in #1.

3. Determine the missing dimensions.

a)

2′ 9�

E

8′ 5�

b)

10.6 cm

12 cm

F

8.5 cm

4. Can you calculate the perimeters of the fi gures in #3? Explain.

Apply It 5. Krista wants to make a simple dress for her niece. She downloads

a dress pattern from the Internet. Krista wants to put a decal above

the waistband on the front. Is there enough room for the decal

shown? Show your work.

Front Back

waistband

14 �12

21�

10�10 �1

2

17 �12

waist


Develop a

Strategy

8′ 5�

- 2′ 9�

What strategy

might you use

to solve this

subtraction

question? Discuss

your strategy with

a peer. Try diff erent

strategies. Do they

produce the same

answer?

Strategy

The patterns used

by seamstresses

and tailors are scale

drawings. To learn

how to adjust a

pattern for another

size, go to www.


learningcentres and

follow the links.

Tools of the Trade

6 �34

6 �34


6. Determine the overall height and the overall length of the stairs

from the fl oor to the landing.

riser = 8�tread = 10�

7. A hospital’s maintenance schedule calls for the walls of all

washrooms to be painted. Below is a scale drawing of one

washroom.

All of the walls are made from concrete block that is 6 inches •

thick.

Th e wall separating the sinks from the toilets goes from the •

fl oor to the ceiling.

Th e ceiling is 9• ′ 6� high.

Th e door is 7• ′ high.

Determine the total area to be painted.

21′

4′

5′

12′

The riser is the

vertical distance

from one step to

another. The tread is

the width of a step

that you walk on.


1. a) Use a scale of 1 square

to 10 cm to create a

scale drawing of the

front of the soccer goal

shown.

b) Use the same scale to

draw the side of the

goal.

2. Bryce plans to renovate his family’s cabin. To help plan, he made a

quick sketch of the layout. Th e diagram is not to scale.

Bedroom 1

20´

18´

8´10´ 6´́

Bedroom 2

a) Th e wall between the bedrooms is made of log and is 8 inches

thick. Determine the missing dimension of bedroom 1 and

of bedroom 2.

b) Draw a scale diagram using the scale 1 square : 1 ft . Th e

bedroom doors are 30 inches wide and 6 inches from the

corner of the room. Th e main door is 36 inches wide and is

centred on the wall. Each window is 36 inches wide and starts

2 feet from the closest corner.

Work With It

2 m2 m

4 m4 m1 m1 m

1.2 m1.2 m

One square foot has

144 square inches.

1 ft2 = 144 in.2


3. A dentist is hiring a local

renovation company to update

the reception area of the offi ce.

a) How many squares make

up 1 metre on the drawing?

b) Count the squares on the

drawing to determine

the dimensions of the

reception area.

c) Determine the area of the fl oor of the reception area.

d) Th e dentist has a lot of riding toys to keep young clients happy

while they are waiting. To protect the walls, the dentist wants

to run carpet along the bottom 10 cm of each wall. How many

square metres of carpet will it take to cover the bottom of

the walls in the reception area?

4. Discuss with a group of peers the strategies used to answer #3d).

Which strategy do you feel the most comfortable with?

5. Th e Newfoundland Tricolour can be of

any size. Th e only condition is that the

ratio of the fl ag’s height to its width must

be 1 : 2. Each colour takes up one third of

the area of the fl ag.

a) Explain the meaning of “a height to width ratio of 1 : 2.”

b) Create a scale model of the fl ag with a height of 15 centimetres.

c) Many tourist shops sell miniature fl ags. On grid paper, or

by using a computer, create a scale model of a fl ag that is

6 inches wide.

6. Make a sketch of an L-shaped room. Include some, but not all, of

the dimensions. Give your sketch to a partner to solve. Have your

partner explain the strategy used for determining the missing

dimensions.

7. A town is considering expanding its

curling facility. Explain why an

architect bidding on the contract

might want to develop scale drawings

or a scale model of the project.

reception area

hall

Scale: 1 square represents 0.5 m

Artists and builders

use scale drawings

to get an accurate

picture of a project

they plan to build.

To learn more about

careers that use

scale drawings, go

to www.mcgrawhill.ca/


and follow the links.

Tools of the Trade

Discuss It

The Newfoundland

Tricolour is a popular

but unoffi cial

provincial fl ag. The

green colour or pale

is always on the

hoist side, which is

the side closest to

the fl agpole.

Draw or

Model

Strategy


Representing Views of

3-D Objects

Construction crews often work from scale drawings. Some

drawings are 2-D views of what the fi nal project will look like.

Other drawings give the impression of being 3-D, even though

they are drawn on paper.

1. Use linking cubes to build a model of a set

of three steps. Th e tread of each step is 40 cm.

Th e riser of each step is 20 cm. Th e steps are

1 m wide. Let one side of each cube

represent 20 cm.

Explore Diff erent Views of a 3-D Object

t

cm.

are

Materials

linking cubes• grid paper • ruler•

Focus On . . .

drawing top, front, and • side views of a given

3-D object

drawing a 3-D object, • given the top, front, and

side views

creating an isometric • drawing of a given

3-D object

determining whether • the given views of a

3-D object represent the

object


2. a) Set the model on a desk. At eye level, look directly at the

front of the steps. What shape do the outside edges of the

model make?

b) Sketch what you see.

c) Set the model on the fl oor. Look directly

at the top of the steps. What shape do the

outside edges of the model make?

d) Sketch what you see.

e) Set the model on a desk. Look directly at the right side of

the model at eye level. What shape do the outside edges of

the model make?

f) Sketch what you see.

3. Refl ect Where would you have to position yourself to have

the model of the steps look like a 3-D object?

4. Extend Your Understanding Make a sketch of just one cube

so that your 2-D drawing gives the impression of being 3-D.

The artist M.C. Escher created a number of works of art that play with

perspective.

Relativity, M.C. Escher

Draw or

Model

Strategy

In this Explore,

you do not need

to draw to scale.

2.2 Representing Views of 3-D Objects • MHR 73

Work With Orthographic DrawingsNicole’s company is bidding on a job to design

and build a garage. Part of Nicole’s presentation

will be a set of orthographic drawings of what

the garage will look like aft er it is built. Th e

dimensions of the garage will be approximately

20 feet long, 12 feet wide, and 12 feet high.

Create orthographic drawings of the garage using a scale of 1 square to

1 foot.

Solution

Method 1: Create Orthographic Drawings by Hand

Front view:

From the front, the garage appears as a rectangle

with a triangular top. You can see the garage door

from this view.

Side view:

From the side, the garage appears as a

longer rectangle. Th e roof also appears as a

rectangle.

side view

Top view:

From the top, only the roof is visible. Since the roof

overhangs the garage a bit, the top view measures

approximately 21 feet by 13 feet. Th e ridge at the

top of the roof is also visible. It is shown by the line

down the centre.

On the Job 1orthographic drawing

a 2-D view of a 3-D • object

often includes a front • view, a top view, and

a side view of the

object

front view

In the side view of an

orthographic drawing,

often only the right

side view is given.

top view


Method 2: Create Orthographic Drawings Using GSP

Open Th e Geometer’s Sketchpad®. Choose Open from the File menu.

Navigate to the directory that stores Th e Geometer’s Sketchpad® on

your computer. Th is will likely be called Sketchpad, and be located

in the Program Files directory. Choose Samples, then Sketches, and

then Geometry. Select and open the fi le Dot Paper.gsp.

Press the button Square dot paper.

You can use the red slider to adjust the spacing of the dots in the pattern.

Select the Straightedge Tool. Use the square dot pattern to help you draw

the front view of the garage. Use the Text Tool to label the drawing.

In the same way, draw the side view and top view of the garage.

Save your fi le with a diff erent name.

Your Turn

Draw and label the front view,

side view, and top view of the

rectangular prism shown. Use

a scale of 1 square to 1 cube.

Draw the prism in two ways:

a) on grid paper by hand

b) using technology

top view

side view

front view


Try It 1. State the dimensions for the front view,

side view, and top view of the rectangular

prism shown.

2. On one sheet of grid paper, draw the

three orthographic views of the prism

in #1. Use a scale of 1 square to 1 metre.

Arrange the drawings on the page

as shown.

3. Redraw the views in #2 using a scale of 1 square to 50 cm.

4. Use technology to create scale drawings of the three views of the

rectangular prism in #1.

5. Sketch the three orthographic views of the set of steps you built in

Explore Diff erent Views of a 3-D Object on page 72. Use a scale of

1 square to 10 cm.

Apply It 6. On paper or using technology, draw a set

of orthographic drawings of the cereal box

shown. Th e dimensions of the box are

approximately 12� by 8� by 3�.

7. Draw the three views of a soup can that is 12 cm high and has a

diameter of 8 cm.

8. A steam room in a fi tness centre measures

10 feet by 6 feet. It has a U-shaped seating

area around three sides of the room. Th e

top view of the seating area is shown.

Sketch the front view and the side view.

Th e height of the seating area is 24 inches.


3 m

4 m

6 mtop

frontside

top view

front view side view


9. A medical CT scanner is shown.

Sketch the front, side, and top

views of the scanner.

10. a) Does each of the three orthographic

drawings represent the

concrete bench shown?

50 cm

50 cm

10 cm

110 cm

side view

183 cm

60 cm

50 cm

front view

183 cm

50 cm

10 cm

top view

b) If not, draw and label the correct view(s).

70 cm70 cm

60 cm

80 cm

60 cm

90 cmA CT scanner is used

to examine patients.

The scanner takes a

series of X-ray views

from many diff erent

angles to produce

cross-sectional

images of the bones

and soft tissues

inside the body.

Patients lie on a bed

that slides inside

the centre hole of

the scanner.

50 cmconcrete bench

50 cm

183 cm

10 cm

110 cm


Work With Isometric DrawingsSometimes people get a better sense of

what an object will look like if a drawing of

it appears 3-D. Photographs have this type

of 3-D eff ect. Th e drawing of the garage

shown is not a photograph, but it gives the

illusion of being 3-D. You can create this

eff ect using an isometric drawing.

How would Nicole create an isometric drawing of the garage?

Solution

Nicole uses isometric dot paper. Her drawing does not include

the slight overhang of the roof.

Step 1 Draw a vertical line to connect

two dots. Th is line is the beginning

of the left side of the garage.

On the Job 2

isometric drawing

a view of a 3-D • object in which

all horizontal edges –

of the object are

drawn at a 30°

angle

all vertical edges –

of the object are

drawn vertically

all lines are drawn –

to scale

For the scale, this

distance between

adjacent dots

represents 2 feet.


Step 2 Connect the bottom of this line to a dot that is at a 30° angle

from it. Th is distance represents 2 feet along the base of the

garage. Extend this line to represent the 12-foot width of

the garage.

Step 3 From the endpoint of this line, draw a line to represent the

20-foot length of the garage.


Step 4 Th e height of the garage, below the roof, is 8 feet. Draw three

lines to represent the height of the garage.

Step 5 Keeping in mind that the distance between adjacent dots

represents 2 feet, complete the diagram of the garage.

Your Turn

Create an isometric drawing of the cube shown.

Web Link

To use an online

isometric drawing

tool, go to www.


learningcentres and

follow the links.


Try It 1. a) Use linking cubes to build a rectangular prism that is

1 cube by 2 cubes by 3 cubes.

b) Use isometric dot paper to create an isometric drawing

of the prism.

2. Create the following shape using

linking cubes. Th en, create an isometric

perspective drawing of it.

3. Create an isometric drawing of the model of the steps that you

built in Explore Diff erent Views of a 3-D Object at the beginning

of this section.

Apply It 4. a) Create any 3-D model you wish using linking cubes.

b) On isometric paper, sketch an isometric drawing of

your model.

5. Draw the front, top, and side views of the

object shown.

6. Create an isometric drawing of the 3-D object shown in the

orthographic drawings. Th e scale is 1 square represents 3 ft .

front view side view top view


Isometric drawing

technology is

used for a variety

of purposes. For

example, software

for designing piping

systems enables

drafters to create

isometric drawings

of piping systems for

industries such as

oil and gas• power generation• chemical• water/waste water• food and beverage• pulp and paper• life sciences•

Tools of the Trade


1. Pete wants to build a shed that is

2 metres wide and 3 metres deep. He

wants the front of the shed to be

2 metres high and the back to be

1.8 metres high. Th e rectangular roof

will be on a slope so that snow can

slide off .

a) Create a set of three orthographic drawings showing the front

view, top view, and right-side view of Pete’s shed.

b) Create an isometric drawing of the shed. Use a scale in which

the distance between vertical dots represents 20 cm.

2. A skateboard ramp is in the shape of a triangular prism.

2 ft

4 ft

6 in.

a) Create orthographic drawings of the front, top, and side views.

Use a scale of your choice.

b) Create an isometric drawing of the ramp. Use a scale of your

choice.

3. Th e legs of a table have a diameter of 10 cm. Th e legs are attached

10 cm from the edges of the tabletop, as shown in the diagram.

Th e table is 150 cm long, 90 cm wide, and 75 cm high.

10 cm10 cm

a) Draw the front view and side view of the whole table.

b) Describe what the top view would look like.

Work With It

1 ft = 12 in.

1

_ 2

ft = 6 in.


4. a) Build a model of the 3-D fi gure that has the following

orthographic views.

top view

front view right side view

b) Create an isometric drawing of the object.

5. Suppose you began an isometric diagram by drawing a circle

on isometric dot paper. Would it look like a sphere when you

completed the drawing? Explain. Include an isometric drawing

with your explanation.

6. Explain why the front view and the side view of this

cylinder look the same.

7. Explain why the isometric

drawing shown cannot

represent a real object.

8. Research and describe at least three jobs that involve working

with scale diagrams.

Discuss It


Representing Perspectives of 3-D Objects

Artists, architects, and interior designers draw 3-D objects in

various ways. The way you draw a 3-D object depends on how

far away you are from the object and what angle you are looking

at it from. How things appear depends on your perspective.

Part 1: The Eff ect of Perspective

on Parallel Lines

1. Go outside and fi nd a pair of

parallel lines, such as the two edges

of a sidewalk. Look down near

your feet and notice the distance

between the parallel lines.

Explore Perspective

Focus On . . .

drawing a one-point • perspective view of a

3-D object

identifying the point of • perspective of a given one-

point perspective drawing

drawing the components of • an exploded view diagram

sketching an exploded • view diagram of a 3-D

object

sketching a 2-D • representation of a 3-D

object, given its exploded

view diagram


2. Refl ect Slowly lift your eyes up and look out along the parallel

lines. What do you notice about how the parallel lines appear as

they get farther away from you?

3. Extend Your Understanding

a) Sketch what you see in step 2.

b) What appears to happen to the parallel lines if they go on

forever?

Part 2: The Eff ect of Perspective on Size

4. a) Stand next to a large object, such as a car or a truck.

b) Stretch your arms out so that you see the entire object

between your outstretched hands.

c) Estimate the distance between your hands.

5. Refl ect

a) Locate a similar car or truck that is farther away from you.

b) Repeat steps 4b) and c). What do you notice?

c) Repeat the steps for a similar object that is even farther away.

What do you notice?

6. Extend Your Understanding

a) Sketch the object from step 4a) on the bottom half of a piece

of paper.

b) Sketch the two objects from step 5 as if they were lined up

behind the fi rst item.

The picture shows a bird on a wall.

a) Without measuring, predict whether

the two red lines are the same length.

b) Measure the two lines. Was your

prediction correct?

c) If your prediction was correct, explain

your reasoning for making that

prediction. If not, explain why you think

you predicted incorrectly.

2.3 Representing Perspectives of 3-D Objects • MHR 85

Work With One-Point Perspectivea) Janet is an artist who is working on a landscape scene. Th e scene

includes a row of shrubs. Janet makes a one-point perspective

drawing of the shrubs.

b) Next, Janet works on a cityscape. She makes a one-point

perspective drawing of a light rail transit train.

Solution

a)

On the Job 1

one-point perspectivedrawing

a drawing in which it • looks like objects in

the background join

at a vanishing point

in the distance

vanishing point

a point toward which • parallel lines appear

to join in the distance

On the left side of her paper,

Janet sketches a shrub.

On the right side of her paper,

Janet sketches a vanishing point.

vanishingpoint

Janet draws a line from the

top of the shrub and from the

bottom of the shrub to the

vanishing point.

Th e lines show Janet the correct

height for drawing the remaining

shrubs so it looks like a row of

shrubs going off into the distance.

The point of perspective of this one-

point perspective drawing is slightly

above and to the right of the original

shrub. How do you know?

point of perspective

the position from • which an object is

being viewed


b)

Your Turn

a) On grid paper, draw a 3-by-3 square in the bottom-left quarter.

b) Choose a vanishing point in the top-right quarter of the page.

c) Draw lines from the top left , top right, and bottom right

corners of the square to the vanishing point.

d) Does the square begin to look like a rectangular prism?

Explain.

When the point of

perspective is

• above an object—

the object appears

to go upward into

the distance

• below an object—

the object appears

to go downward

into the distance

• to the right of an

object—the object

appears to go to

the right into the

distance

• to the left of an

object—the object

appears to go to

the left into the

distance

Point of perspective:up and to the left

Point of perspective:down and to the right

On the right side of her

paper, Janet sketches the face

of a train.

On the left side of her paper,

Janet sketches a vanishing

point.

Janet draws a line from the

top and from the bottom of

the face of the train.

Th e lines show Janet how

to draw the rest of the train

so it looks like it is going off

into the distance.

vanishingpoint

What is the point of perspective of

the drawing?


Try It 1. a) Build a model of a rectangular prism with linking cubes.

b) On graph paper, sketch a scale diagram of one rectangular

face of the prism in the bottom-left quarter of the page.

c) Locate a vanishing point on the right side of the page near

the middle. Create a one-point perspective drawing of the

rectangle.

d) Does the drawing appear to represent your model of a

rectangular prism?

2. a) In the middle of a piece of paper, copy the

shape shown.

b) Mark a vanishing point on the left side of

the paper. Create a one-point perspective

drawing of the shape.

c) What is the point of perspective of the

drawing? How do you know?

3. Th e drawing shows a one-point perspective

drawing.

a) Identify the shape.

b) What is the point of perspective of the

drawing?

c) Using the same shape, create a one-point perspective

drawing from a diff erent point of perspective.

d) What is the point of perspective of your drawing?

4. a) Draw a triangle in the middle of a sheet of paper.

b) Create a vanishing point on the paper.

c) Sketch a one-point perspective drawing.

d) Which 3-D fi gure does the drawing appear to represent?

e) What is the point of perspective of the drawing?



5. Identify the point of perspective for each drawing.

a)

b)

c) d)

Apply It 6. a) On grid paper, draw a 4-by-4 square in the bottom-right quarter

of the page. Place a vanishing point in the upper-left quarter.

Create a one-point perspective drawing of a rectangular prism.

b) On another sheet of grid paper, draw a 4-by-4 square in the

bottom half of the page near the middle. Place a vanishing

point above the square in the top half of the page. Create a

one-point perspective drawing of a rectangular prism.

c) Compare the two drawings. Does changing the location of

the vanishing point give you a diff erent impression of the

rectangular prism? Explain.

7. a) On the left side of a piece of paper, sketch any object.

b) On the right side of the paper, place a vanishing point.

c) Draw lines from the top of the object and from the bottom

of the object to the vanishing point.

d) Using the lines as a guide, draw a vanishing row of the object

you drew.


8. a) On grid paper, near the centre of the page, write your fi rst

name in block letters. For example, a J might appear as shown.

b) Locate a vanishing point an inch or so above the midpoint of

your name.

c) Create a one-point perspective drawing of your name.

d) What is the point of perspective of your drawing?

9. Create four one-point perspective drawings of a 3-D object of your

choice. Use a diff erent point of perspective for each drawing.

10. a) Create a drawing or painting of an outdoor scene. Include at

least three objects drawn from a one-point perspective.

b) Show your artwork to a classmate. Discuss the 3-D eff ect in

each other’s artwork.

11. a) Draw a four-sided 2-D shape.

b) Create an isometric drawing of the shape.

c) Create a one-point perspective drawing of the shape.

d) What are the similarities and diff erences between your

drawings in part b) and part c)?

The picture shows a representation of a 3-D fi gure. Identify the total

number of sides of the fi gure.


Work With Exploded View DiagramsGeorge writes instruction sheets for how to assemble “build-it-

yourself ” furniture. His instructions include a list of parts, their

quantity, and the steps needed to build the piece of furniture. He

also includes an exploded view diagram. George’s completed

instructions are always done using drawing soft ware, but he oft en

starts by making sketches by hand.

Currently, George is working on the assembly instructions for

putting the legs on a coff ee table.

a) List the steps for assembling the table.

b) Create an exploded view diagram of the assembly.

Solution

a) Follow these steps:

Step A Place the tabletop face down on the fl oor.

Step B Line up the holes in the plate with the predrilled holes

in the underside of the tabletop.

Step C Insert four screws to attach the plate to the underside

of the tabletop.

Step D Screw the leg of the table into the threaded hole in

the plate.

Step E Repeat steps B, C, and D for the other three legs.

On the Job 2

exploded view diagram

a drawing that shows • the components

of an object with

the parts slightly

separated

often used to show • the sequence of steps

for assembling an

object

usually shown as • a set of isometric

drawings

Make a

Systematic

List

Strategy


b)

A

B – 4×

C – 16×

D – 4×

Your Turn

Th e end table that matches the coff ee table does not use plates.

Each of the four legs screws directly into a predrilled hole in the

underside of the table. Sketch an exploded view diagram that

shows the assembly of one leg into the end table.

The diagram shows an exploded view of the pieces of a 3-D puzzle. Create

a colour isometric drawing of the completed puzzle.


Try It 1. a) Identify the item shown in the exploded view diagrams.

b) How many parts are shown in the diagram below?

2. a) Identify the item shown in the exploded view diagram.

b) What names would you give to parts B, D, M, and T?

A

B

C

D

E

H

I JK

M

P

R

S

Q

N

OL

G

F

T



3. a) Using linking cubes, build a rectangular prism that is 3 by 1

by 1.

b) Separate the cubes. Line them up in the order you assembled

them in part a).

c) On isometric dot paper, sketch an exploded view diagram of

the rectangular prism. Make it a scale diagram. Label your

diagram with the scale you used.

Apply It 4. a) List the ingredients that someone would use to make a peanut

butter and jam sandwich.j

b) Starting with the bottom piece of bread, list the ingredients

in the order that the person would build the sandwich.

c) Create and label an exploded view diagram of the sandwich.

5. Choose an item in your classroom or school, such as a chair,

that is made up of a number of parts.

a) List the parts and the quantities of each part that make up

the item.

b) Write the sequence of steps for assembling the item.

c) Draw an exploded view diagram of the assembly of the item.

6. Th e exploded view diagram shows

a birdhouse. On grid paper, sketch

the front view and the side view of

the birdhouse.

Fast food chains

often use exploded

view diagrams of

food items, such as

sandwiches. They

use these drawings

to train employees

and to ensure that

the food items are

made exactly the

same way in every

location.

8�

4�

4�6�

4�

6�

4�


1. a) On grid paper, sketch a rectangle near the centre of the page.

Th is rectangle represents the front of a bus.

b) Locate a vanishing point near the bottom-left corner of the

page. Create a one-point perspective drawing so it looks like

a bus coming toward you.

c) As the viewer of the bus, what is your point of perspective?

d) Create a similar diagram, this time locating the vanishing

point near the top-left corner of the page.

e) As the viewer of the bus, what is your point of perspective

now?

2. Create and label an exploded

view diagram of a double

cheeseburger. Include the

condiments of your choice.

3. A high school runs an annual plant sale in the spring. Th ey plan to

advertise with fl yers. Design the heading for the fl yer by making a

one-point perspective drawing of the words PLANT SALE.

4. a) Create a product logo of your choice that appears 3-D.

b) Explain how you created the eff ect.

5. Describe a specifi c situation in which you might choose to

create an exploded view diagram.

6. How does the location of the vanishing point aff ect the point

of perspective for a one-point perspective diagram?

Work With It

Discuss It


Working With Scale, pages 60–71

1. Determine

the missing

dimensions.

2. Danny’s toy tractor is a 1 : 16 scale model of the actual tractor.

a) Th e wheelbase of the toy tractor is 16 cm. How long is the

wheelbase of the real tractor?

b) Th e front tread range of the real tractor is 60 in. What is the

front tread range of the toy tractor?

2.1

13 m11 m

6 m

1.5 mA B

C1 m2 m

The wheelbase of

a vehicle is the

distance between

the front axle and

the rear axle. The

tread range is the

distance between

the centre of the

tread of one of the

tires to the other.

tread range

wheelbase

Section After this section, I know how to . . .

2.1 use proportional reasoning to determine the dimensions of a scale drawing or model

draw a scale diagram of a given object

solve problems that involve scale

construct a model of a 3-D object

2.2 draw top, front, and side views of a given 3-D object

draw a 3-D object, given the top, front, and side views

create an isometric drawing of a given 3-D object

determine whether the given views of a 3-D object represent the object

2.3 draw a one-point perspective view of a 3-D object

identify the point of perspective of a given one-point perspective drawing

draw the components of an exploded view diagram

sketch an exploded view diagram of a 3-D object

sketch a 2-D representation of a 3-D object, given its exploded view diagram

What You Need to Know

If you are unsure about any of these questions, review the appropriate

section or sections of this chapter.


Representing Views of 3-D Objects, pages 72–83

3. Th e object shown is made from linking cubes. On paper or

using technology, create a set of three orthographic drawings

of the object.

4. Th e picture shows a set of orthographic drawings of a 3-D object.

front view

top view right side view

a) Use linking cubes to build a model of the object.

b) On isometric dot paper or using technology, create an

isometric drawing of the object.

Representing Perspectives of 3-D Objects, pages 84–95

5. On paper or using technology, draw your initials in large block

letters. Create a one-point perspective drawing of your initials.

6. Th e picture shows a set of Allen keys.

Th e keys fi t into the green arms, which

fi t into the centre piece. On paper or

using technology, draw an exploded

view diagram of the Allen key.

2.2

2.3

An Allen key,

sometimes called

an Allen wrench,

is a tool used to

loosen or tighten

screws and bolts

with hexagonal

slots. Some models

of “build-it-yourself”

furniture require an

Allen key.

Tools of the Trade

Skill Check • MHR 97

Test Yourself

For #1 to #5, select the best answer.

1. What kind of drawing is this diagram?

A exploded view drawing

B isometric drawing

C one-point perspective drawing

D orthographic drawing



B isometric drawing





B isometric drawing





B isometric drawing





B isometric drawing




6. Draw the front view, top view, and side view

of the object shown here.

7. Create an isometric drawing of the object shown in the orthographic

diagrams.

front

top right side

8. a) Create a one-point perspective drawing of a triangular prism.

b) Identify the point of perspective of your drawing.

9.

a) Write the sequence of steps for installing the light switch cover.

b) Sketch an exploded view diagram for installing the light switch cover.

Test Yourself • MHR 99

Project Design Your Own Model

You are now ready to design your own model.

1. Choose something you would like

to design. It can be as simple or as

complex as you like. For example:

You want to build a model or a �

piece of scenery for a train set, a

miniature dollhouse, or a stage

performance.

You wish to design and create an �

article of clothing for a collectible

doll or for a sports mascot.

You plan to make money selling �

craft s during the summer. What

miniature would remind tourists

of where you live? For example,

in Newfoundland and Labrador,

they might buy a miniature fl ag,

a small replica of Signal Hill, or a

fridge magnet in the shape of the province.

2. a) Choose a scale for your design.

b) Draw your design using two of the following:

a set of orthographic drawings•

a one-point perspective drawing•

an isometric drawing•

an exploded view drawing•

3. a) Include a list of the materials that you will need in order to

build the model.

b) Estimate the cost of these materials.

4. Build all or part of your design.

Chapter Project


G A M E S A N D

P U Z Z L E S

Model It

1. a) How many linking cubes would you need to build the

3-D sculpture shown in the four diagrams below?

b) Use linking cubes to build the 3-D sculpture represented

in the four diagrams.

front view

side view

top view

back view

2. a) Use linking cubes to build an original 3-D object.

(Keep it hidden from most of the class.)

b) Draw the front, back, side, and top views of the sculpture.

c) Trade views with a partner. Draw the views of each other’s

objects.

d) Assess the orthographic views drawn by your partner.

Where do you agree with the views?•

Where do you disagree?•

Work with your partner to develop a correct set of orthographic •

views for your 3-D sculpture.

Materials

linking cubes• grid paper • ruler•

Games and Puzzles • MHR 101

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